SLIDE 1
New examples of matrix orthogonal polynomials satisfying second order differential equations. Structural formulas
Joint work with A.J. Dur´ an
University of
Doc-course,University of Seville, spring 2010
SLIDE 2 Introduction
Given a self-adjoint positive definite matrix valued weight function W(t) (of dimension N × N) consider the skew symmetric bilinear form defined for any pair of matrix valued functions P(t) and Q(t) by the numerical matrix P, Q = P, QW =
P(t)W(t)Q∗(t)dt, where Q∗(t) denotes the conjugate transpose of Q(t). There exists a sequence (Pn)n of matrix polynomials, orthonormal with respect to W and with Pn of degree n. The sequence (Pn)n is unique up to a product with a unitary matrix.
SLIDE 3
INTRODUCTION
Property Any sequence of orthonormal matrix valued polynomials (Pn)n satisfies a three term recurrence relation A∗
nPn−1(t) + BnPn(t) + An+1Pn+1(t) = tPn(t),
where P−1 is the zero matrix and P0 is non singular. An are nonsingular matrices and Bn hermitian. Considering possible applications of MOP it is natural to concentrate on those cases where some extra property holds.
SLIDE 4 The Matrix Bochner’s Theorem
In the nineties, A. Dur´ an takes a step in this direction raising the problem of characterizing MOP which satisfy second order differential equations. Dur´ an, Rocky Mountain J. Math (1997) Characterize all families of MOP satisfying Pnℓ2,R = P
′′
n F2(t) + P
′
nF1(t) + PnF0(t) = ΛnPn(t), n ≥ 0
Right hand side differential operator ℓ2,R = D2F2(t) + D1F1(t) + D0F0(t). Pn eigenfunctions, Λn eigenvalues: Pnℓ2,R = ΛnPn
SLIDE 5
The matrix Bochner Problem
The first examples of MOP, which does not reduce to scalar, satisfying 2nd order differential equations in the framework of the general theory of orthogonal polynomials appeared in Dur´ an-Gr¨ unbaum , Orthogonal Matrix Polynomials satisfying differential equations Int. Math Res. Not. 2004.
SLIDE 6
The goal
We aim to: Present a new example of MOP, of dimension N × N, satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as
Rodrigues formula Recurrence relations
SLIDE 7 MOP related to second order differential op.
Consider the recurrence relation for the sequence MOP (Pn(t))n≥0,
- rthonormal with respect to a weight matrix W
A∗
nPn−1(t) + BnPn(t) + An+1Pn+1(t) = tPn(t),
where P−1 is the zero matrix and P0 is non singular. The examples of MOP satisfying second order differential equations existing up to now in the literature satisfy that lim
n→∞
An ϕ(n) = aI, lim
n→∞
Bn ϕ(n) = bI, for a convenient continuous function ϕ(t). The example given here satisfies the property that the previous limits give no longer a scalar matrix
SLIDE 8 The method to find MOP satisfying 2nd order differential eq.s
(Dur´ n-Gr¨ unbaum), 2004 Simmetry Eqs: differential equations for the weight function W and the coefficients of ℓ2,R = D2F2(t) + D1F1(t) + D0F0(t). Symmetry Equations F2W = WF ∗
2
2(F2W)′ = F1W + WF ∗
1
(F2W)
′′ − (F1W)′
= (WF ∗
0 − F0W)
We put W(t) = T(t)T ∗(t)
SLIDE 9 The talk continues according to the following suggestions: To carefully present the N × N example and the differential
- equation. Then to show the 2 × 2 case. Mention the method
follow to obtain the generalization (the Theorem 2.3 of the paper [A. Dur´ an, Constructive Approx, 2009]) To present the Rodrigues formula, mentioning how it was
- btained, that is, the differential equation for Rn which
appeared in [A. Dur´ an, Int. Math. Research Notices, 2010] Finally the recurrence relation for the simplest normalization and the recurrence for the orthonormal polynomials, remarking the facts that the limits of the recurrence coefficients are not scalar.
